Subgroups — #37576211

Order 24448.

508 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259322	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20259281	C₂	1	1	yes	maximal, characteristic
3	#27132586	C₂	1	2	yes	maximal
4	#27132575	C₂	1	2	yes	maximal
5	#27132567	C₂	1	2	yes	maximal, complemented
6	#27132561	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132561	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132550	C₂	1	2	yes	maximal, complemented
9	#27132529	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132523	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
13	#20259322	—	2	4	no	—
14	#20259281	—	2	4	no	—
15	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
25	C₁₉₁⋊#95	C₂²	1	2	yes	—
26	C₁₉₁⋊#95	C₂²	1	2	yes	—
27	C₁₉₁⋊#95	C₂²	1	2	yes	—
28	C₁₉₁⋊#95	C₂²	1	2	yes	—
29	C₁₉₁⋊#95	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
35	#95×C₁₉₁	C₂²	1	1	yes	characteristic
36	#95×C₁₉₁	C₂²	1	1	yes	characteristic
37	C₂×Dic₃₈₂	—	2	4	no	—
38	C₂×Dic₃₈₂	—	2	4	no	—
39	C₂×Dic₃₈₂	—	4	2	no	—
40	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
44	C₄⋊Dic₁₉₁	—	2	4	no	—
45	C₄⋊Dic₁₉₁	—	2	2	no	—
46	C₄⋊Dic₁₉₁	—	2	2	no	—
47	C₄⋊Dic₁₉₁	—	2	2	no	—
48	C₄⋊Dic₁₉₁	D₄	1	4	yes	—
49	Dic₁₉₁⋊C₄	—	2	4	no	—
50	Dic₁₉₁⋊C₄	—	4	2	no	—
51	C₂²⋊Dic₁₉₁	—	2	4	no	—
52	C₂²⋊Dic₁₉₁	—	4	2	no	—
53	C₄×Dic₁₉₁	—	2	4	no	—
54	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₉₁	—	2	2	no	—
56	C₂²×C₄×C₁₉₁	—	2	2	no	—
57	C₂²×C₄×C₁₉₁	Q₈	1	4	yes	complemented
58	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
64	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
65	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
66	C₄²×C₁₉₁	—	2	2	no	—
67	C₄²×C₁₉₁	—	2	2	no	—
68	Dic₃₈₂	—	4	8	no	—
69	C₂×Dic₁₉₁	—	2	2	no	—
70	C₂×Dic₁₉₁	—	2	2	no	—
71	C₂×Dic₁₉₁	—	2	2	no	—
72	C₂×Dic₁₉₁	—	2	2	no	—
73	C₂×Dic₁₉₁	—	2	2	no	—
74	C₂×Dic₁₉₁	—	2	2	no	—
75	C₂×Dic₁₉₁	—	2	2	no	—
76	C₂×Dic₁₉₁	—	2	2	no	—
77	C₂×Dic₁₉₁	—	2	2	no	—
78	C₂×Dic₁₉₁	—	2	2	no	—
79	C₂×Dic₁₉₁	—	2	2	no	—
80	C₂×Dic₁₉₁	C₂×D₄	1	4	yes	—
81	C₂³×C₁₉₁	—	2	1	no	—
82	C₂³×C₁₉₁	—	2	1	no	—
83	C₂³×C₁₉₁	—	2	1	no	—
84	C₂³×C₁₉₁	—	2	1	no	—
85	C₂³×C₁₉₁	—	2	1	no	—
86	C₂³×C₁₉₁	—	2	1	no	—
87	C₂³×C₁₉₁	C₂×Q₈	1	2	yes	—
88	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
89	C₂×C₄×C₁₉₁	—	2	1	no	—
90	C₂×C₄×C₁₉₁	—	2	1	no	—
91	C₂×C₄×C₁₉₁	—	2	1	no	—
92	C₂×C₄×C₁₉₁	—	2	1	no	—
93	C₂×C₄×C₁₉₁	—	2	1	no	—
94	C₂×C₄×C₁₉₁	—	2	1	no	—
95	C₂×C₄×C₁₉₁	—	2	8	no	—
96	C₂×C₄×C₁₉₁	—	2	1	no	—
97	C₂×C₄×C₁₉₁	—	2	1	no	—
98	C₂×C₄×C₁₉₁	—	2	1	no	—
99	C₂×C₄×C₁₉₁	—	2	1	no	—
100	C₂×C₄×C₁₉₁	—	2	1	no	—
101	C₂×C₄×C₁₉₁	—	2	1	no	—
102	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
104	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
105	C₂×C₄×C₁₉₁	#41	1	2	yes	—
106	C₂×C₄×C₁₉₁	#41	1	2	yes	—
107	C₂×C₄×C₁₉₁	#41	1	2	yes	—
108	C₂²×C₁₉₁	—	2	2	no	—
109	C₂²×C₁₉₁	—	2	2	no	—
110	C₂²×C₁₉₁	—	2	2	no	—
111	C₂²×C₁₉₁	—	4	1	no	—
112	C₂²×C₁₉₁	—	4	1	no	—
113	C₂²×C₁₉₁	—	4	1	no	—
114	C₂²×C₁₉₁	—	4	1	no	—
115	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
117	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
118	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
119	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
122	C₄×C₁₉₁	—	2	2	no	—
123	C₄×C₁₉₁	—	2	2	no	—
124	C₄×C₁₉₁	—	2	2	no	—
125	C₄×C₁₉₁	—	4	1	no	—
126	C₄×C₁₉₁	—	4	1	no	—
127	C₄×C₁₉₁	C₂²⋊Q₈	1	4	yes	—
128	Dic₁₉₁	—	2	4	no	—
129	Dic₁₉₁	—	4	2	no	—
130	Dic₁₉₁	—	4	2	no	—
131	Dic₁₉₁	—	4	2	no	—
132	C₂×C₁₉₁	—	4	2	no	—
133	C₂×C₁₉₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₁₉₁	#533	1	1	yes	characteristic
135	C₂×C₁₉₁	#544	1	1	yes	characteristic
136	C₂×C₁₉₁	#546	1	1	yes	characteristic
137	C₂×C₁₉₁	#548	1	1	yes	characteristic
138	C₂×C₁₉₁	#555	1	1	yes	characteristic
139	C₂×C₁₉₁	#556	1	1	yes	characteristic
140	C₁₉₁	#2439	1	1	yes	characteristic, complemented, Sylow
141	#2439	—	191	1	no	maximal, Sylow
142	C₂×C₄⋊Q₈	—	191	1	no	—
143	C₂×#117	—	191	1	no	—
144	#396	—	191	2	no	—
145	#393	—	191	2	no	—
146	#391	—	191	2	no	—
147	#389	—	191	1	no	—
148	#389	—	191	1	no	—
149	#387	—	191	2	no	—
150	#382	—	191	1	no	—
151	#380	—	191	1	no	—
152	C₄×C₂²⋊C₄	D₁₉₁	1	1	yes	characteristic
153	C₂²×Q₈	—	191	2	no	—
154	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
155	C₄⋊Q₈	—	382	4	no	—
156	#117	—	382	4	no	—
157	C₂×C₄⋊C₄	—	191	1	no	—
158	C₂×C₄⋊C₄	—	191	1	no	—
159	C₂×C₄⋊C₄	—	191	2	no	—
160	C₂×C₄⋊C₄	—	191	1	no	—
161	C₂×C₄⋊C₄	—	191	2	no	—
162	C₂×C₄⋊C₄	—	191	1	no	—
163	C₂×C₄⋊C₄	—	191	2	no	—
164	C₂×C₂²⋊C₄	—	191	2	no	—
165	C₂×C₂²⋊C₄	—	191	2	no	—
166	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
168	C₂×C₄²	—	191	2	no	—
169	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
170	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
171	#95	—	191	2	no	—
172	#95	—	191	2	no	—
173	#95	—	191	2	no	—
174	#95	—	191	2	no	—
175	#95	—	191	2	no	—
176	#95	C₂×D₁₉₁	1	1	yes	characteristic
177	#95	C₂×D₁₉₁	1	1	yes	characteristic
178	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
179	C₂×Q₈	—	382	4	no	—
180	C₂×Q₈	—	382	4	no	—
181	C₂×Q₈	—	764	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	191	2	no	—
185	C₂²×C₄	—	191	2	no	—
186	C₂²×C₄	—	191	2	no	—
187	C₂²×C₄	—	191	2	no	—
188	C₂²×C₄	Dic₃₈₂	1	4	yes	complemented
189	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
195	C₄⋊C₄	—	382	4	no	—
196	C₄⋊C₄	—	382	2	no	—
197	C₄⋊C₄	—	382	2	no	—
198	C₄⋊C₄	—	382	2	no	—
199	C₄⋊C₄	—	382	4	no	—
200	C₄⋊C₄	—	191	4	no	—
201	C₄⋊C₄	—	764	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	382	4	no	—
205	C₂²⋊C₄	—	764	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	382	4	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	C₂×Dic₃₈₂	1	2	yes	—
216	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
217	Q₈	—	764	8	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	382	2	no	—
232	C₂×C₄	—	382	2	no	—
233	C₂×C₄	—	382	2	no	—
234	C₂×C₄	—	382	2	no	—
235	C₂×C₄	—	382	2	no	—
236	C₂×C₄	—	382	2	no	—
237	C₂×C₄	—	382	2	no	—
238	C₂×C₄	—	382	2	no	—
239	C₂×C₄	—	382	2	no	—
240	C₂×C₄	—	382	2	no	—
241	C₂×C₄	—	382	2	no	—
242	C₂×C₄	—	191	4	no	—
243	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
244	C₂×C₄	C₂×Dic₃₈₂	1	2	yes	—
245	C₂×C₄	C₂×Dic₃₈₂	1	2	yes	—
246	C₂×C₄	#15690286	1	2	yes	—
247	C₂×C₄	#15690287	1	2	yes	—
248	C₂×C₄	#15690287	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
257	C₂²	C₂²×Dic₃₈₂	1	1	yes	characteristic
258	C₂²	C₂×#15690286	1	1	yes	characteristic
259	C₂²	C₂×#15690287	1	1	yes	characteristic
260	C₂²	C₂×#15690287	1	1	yes	characteristic
261	C₂²	#20259368	1	1	yes	characteristic
262	C₂²	#20259369	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	—	382	4	no	—
269	C₄	—	764	2	no	—
270	C₄	—	764	2	no	—
271	C₄	—	764	2	no	—
272	C₄	#20259299	1	4	yes	—
273	C₂	—	4	2	no	—
274	C₂	C₂×#20259299	1	1	yes	characteristic
275	C₂	#27133220	1	1	yes	characteristic
276	C₂	#27133291	1	1	yes	characteristic
277	C₂	#27133309	1	1	yes	characteristic
278	C₂	#27133322	1	1	yes	characteristic
279	C₂	#27133378	1	1	yes	characteristic
280	C₂	#27133383	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator, socle, cosocle	C₂³×C₁₉₁	C₂⁴
Fitting	C₄×C₂²⋊C₄×C₁₉₁	C₂
prime radical	C₂⁴×C₁₉₁	C₂³
hypercenter	C₄×C₂²⋊C₄	D₁₉₁
hypocenter	C₁₉₁	#2439
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#20259322	C₂	semidirect
5. #27132567	C₂	semidirect
6. #27132561	C₂	semidirect
7. #27132561	C₂	semidirect
8. #27132550	C₂	semidirect
9. #27132529	C₂	semidirect
57. C₂²×C₄×C₁₉₁	Q₈	semidirect
140. C₁₉₁	#2439	semidirect
188. C₂²×C₄	Dic₃₈₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#20259322	2
C₂×#20259281	2
#27132586	2
#27132575	2
#27132567	2
#27132561	2
#27132550	2
#27132529	2
#27132523	2
C₄×C₂²⋊C₄×C₁₉₁	2
#2439	191